Let $(f_n)_{n\ge0}$ be a real sequence. Then $\sum f_n {x^n \over n!}$ is called the exponential generating function of $(f_n)$.

Let $k\ge0$ be a nonnegative integer. If we add another factorial $(n+k)!$ to the denominator and obtain $$\sum_{n\ge0} f_n {x^n \over n!(n+k)!} , $$ is there a name for this kind of generating functions? Perhaps "Bessel generating function"?

  • $\begingroup$ May I ask why you are interested in giving your series a name? I would be tempted to call it the generating function of the sequence $(f_n/(n+k)!)$. $\endgroup$ Oct 13, 2014 at 11:01
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    $\begingroup$ @JoonasIlmavirta Thanks for your comment. Because it might not be "my" series. I am quite certain that examples of this series have been studied before, maybe even some general theories exist. So if I know the name then I can search for related literature. $\endgroup$
    – Y. Pei
    Oct 13, 2014 at 11:05
  • $\begingroup$ I guess the theory of such series relies on the bivariate egf $$\sum_{n\ge0, m\ge0} f_n [m\ge n] \frac{x^n}{n!}\frac{y^m}{m!}.$$ (here $[m\ge n]$ is $1$ or $0$ according whether $m\ge n$ or not). $\endgroup$ Oct 13, 2014 at 11:45

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No, there is no name for this kind of generating function, except in the case $k=0$, when they are called “doubly exponential generating functions”. I do not know of any applications for $k>0$.

  • $\begingroup$ I guess the case $k>0$ arises naturally as the $k$th derivative of a doubly exponential generating function. This would seem to reduce the study of "Bessel" generating series to that of doubly exponential ones. $\endgroup$ Oct 13, 2014 at 14:04
  • $\begingroup$ It's unlikely that these will have any combinatorial applications since the product of two "Bessel generating functions" with integer coefficients won't in general have integer coefficients (by which I mean the coefficients of $x^n/n!(n+k)!$). $\endgroup$
    – Ira Gessel
    Oct 13, 2014 at 17:12

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